Lipschitz functions on spaces of homogeneous type

✍ Scribed by Roberto A Macías; Carlos Segovia

• Publisher: Elsevier Science
• Year: 1979
• Tongue: English
• Weight: 552 KB
• Volume: 33
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1016/0001-8708(79)90012-4

✦ Synopsis

Lipschitz Functions on Spaces of Homogeneous Type

Results on the geometric structure of spaces of homogeneous type are obtained and applied to show the equivalence of certain classes of Lipschitz functions defined on these spaces.

I. YOTATION AND DEFINITIONS

By a quasi-distance on a set X we mean a non-negative function d(x, y) defined on S x X, such that (i) for every x' and y in S, d(s, y) = 0 if and only if s = y, (ii) for every .v and y in X, d(~, y) = d(y, X) and (iii) there exists a finite constant K such that for every X, y and z in S d(x, y) ,( q+, 2) + d(z, y)).

A quasi-distance d(x, y) defines a uniform structure on X. The balls B(x, r) = {y: 4% Y) < 4, Y > 0, form a basis of neighbourhoods of w for the topology induced by the uniformity on ,Y. This topology is a metric one since the uniform structure associated to d(~, y) has a countable basis. We shall refer to this topology as the d-topology of X. We say that two quasi-distances d(x, y) and d'(~, y) on S are equivalent if there exist two positive and finite constants, c1 and ca , such that c&x, y) < d'(~, y) < c&(x, y) hold for every x and y in S. We observe that the uniformities and the topologies defined by equivalent quasidistances coincide.

Let X be a set endowed with a quasi-distance d(.v, y) and assume that a positive measure CL, defined on a a-algebra of subsets of X which contains the d-open subsets and the balls B(x, Y), is given and satisfies that there exist two finite constants, a > 1 and A, such that 0 < p(B(x, UT)) < A .